p-harmonic coordinates for H\"older metrics and applications

TL;DR

This paper establishes the existence of p-harmonic coordinate systems on manifolds with H"older continuous metrics and applies this to regularity and conformal flatness results, extending previous work to less regular metrics.

Contribution

It introduces p-harmonic coordinates for H"older metrics and applies them to regularity and conformal geometry, extending prior results to lower regularity settings.

Findings

Existence of p-harmonic coordinates on H"older manifolds

Regularity of conformal maps between H"older manifolds

Conditions for local conformal flatness with low regularity metrics

Abstract

We show that on any Riemannian manifold with H\"older continuous metric tensor, there exists a $p$-harmonic coordinate system near any point. When $p = n$ this leads to a useful gauge condition for regularity results in conformal geometry. As applications, we show that any conformal mapping between manifolds having $C^\alpha$ metric tensors is $C^{1+\alpha}$ regular, and that a manifold with $W^{1,n} \cap C^\alpha$ metric tensor and with vanishing Weyl tensor is locally conformally flat if $n \geq 4$. The results extend the works [LS14, LS15] from the case of $C^{1+\alpha}$ metrics to the H\"older continuous case. In an appendix, we also develop some regularity results for overdetermined elliptic systems in divergence form.